
theorem
  for V being finite-dimensional RealUnitarySpace, W1,W2 being Subspace
  of V st V is_the_direct_sum_of W1, W2 holds dim V = dim W1 + dim W2
proof
  let V be finite-dimensional RealUnitarySpace;
  let W1,W2 be Subspace of V;
  assume
A1: V is_the_direct_sum_of W1, W2;
  then
A2: the UNITSTR of V = W1 + W2 by RUSUB_2:def 4;
  W1 /\ W2 = (0).V by A1,RUSUB_2:def 4;
  then (Omega).(W1 /\ W2) = (0).V by RUSUB_1:def 3
    .= (0).(W1 /\ W2) by RUSUB_1:30;
  then dim(W1 /\ W2) = 0 by Th12;
  then dim W1 + dim W2 = dim(W1 + W2) + 0 by Th15
    .= dim (Omega).V by A2,RUSUB_1:def 3
    .= dim V by Th10;
  hence thesis;
end;
